Attending two semesters in the US at different Universities without a two year stay back home [closed]

This is my current situation:

I plan to study a semester abroad in the US (I'm German) at Drexel University from ~March to ~June 2016. This is a normal, exchange-like program between my university and Drexel.

Additionally, I have been given the opportunity to write my bachelors thesis at a research group at Yale University, which I plan to do right after my stay at Drexel. This allows me to save the costs for another flight and fills my German six-to-seven-month-semester nicely.

A few days ago, I realized that the visa may become a problem, as I know from people who have been working at Yale (at that exact research group), they will (typically?) issue a J-1 visa. A brief research revealed that Drexel will probably issue a J-1 visa as well. Drexel can, afaik, also issue F-1 visas, but that requires me to "be enrolled in a program or course of study thatculminates in a degree, diplomaor certificate". (Found here, p. 4)

As I also learned today, a J-1 visa requires me to return to Germany and not re-enter the US for two years after the program ends.

Now the question is this: What are my options to get all this visa-stuff right and do both, study at Drexel and write my bachelors thesis at Yale?

Attempt a School Transfer from Drexel to Yale? (I only found information about that in the Drexel student handbook, same page as cited above)
Attempt to get a J-1 visa from either of the two institutions for the whole duration of my stay (note that a warning about entering on a J-1 visa with a different institution named on it than the one where you plan to study can be found in the student handbook)
Attempt to have the work at Yale classified as Academic Training for J-1 Students by Drexel
Somehow try to get an F-1 visa and have the work at Yale classified as either Curricular Practical Training or Optional Practical Training by Drexel (cannot post links, not enough reputation, find it on the same level as the Academic Training for J-1 Students)
Do something else?

Note that I have to return to Germany to finish my study, as I'm still missing some ECTS.

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Can you be enrolled in two universities at once USA?

In many cases, the surprising answer is that you can. You may have heard the term "dual enrollment," which sometimes refers to high school students who are also taking classes at a local college. However, dual enrollment can also refer to students attending two colleges.

Can international students take a gap year?

A gap year provides a chance for international students to do something significant that does more than just \u201cpad\u201d their resume. As the Harvard authors explain: Occasionally students are admitted to Harvard or other colleges in part because they accomplished something unusual during a year off.

Can international students take hybrid classes?

Federal officials extended a pandemic-era policy that allows international students to take online and hybrid courses at American colleges and universities through the 2022-23 academic year on Monday.

Can F-1 students take hybrid courses?

As of Fall 2021, the International Center is interpreting COVID-19 guidelines to still apply. F-1 or J-1 students arriving on campus may continue to take remote or online classes, but should include at least one in-person or hybrid class in their schedules.

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